feedbacks when the studied system possesses a property of robust stabilizability by bounded state feedback.
The chapter is organized as follows. Some notation and definitions are proposed in Section 6.2. A continuous-discrete interval observer is presented in Section 6.3. Section 6.4 is devoted to an illustrative example. Conclusions are drawn in Section 6.5.
over every interval Ii, by
˙
x(t) = Ax(t) +Bu(t) +δ1(t),
y(t) = Cx(ti) +δ2,i, ∀t∈Ii , (6.2) wherex∈Rn are the state variables,u: [0,+∞)→Rp is the input,y∈Rq is the output, δ1 : [0,+∞)→Rnis a disturbance, which is supposed to be piecewise continuous function, δ2,i∈Rq is a disturbance and A∈Rn×n,B ∈Rn×p andC ∈Rq×n are constant matrices.
We introduce three assumptions.
Assumption 6.1. There is a matrixL∈Rn×q such that the spectral radius of the matrix
G=J eνA, (6.3)
with
J =In−LC, (6.4)
is smaller than1, i.e. Gis Schur stable. Moreover there exists an invertible matrixP ∈Rn×n such that the matrix G=P GP−1 is nonnegative.
Assumption 6.2. There is a Lipschitz continuous feedback us satisfying us(0) = 0 such that, for every functiond: [0,+∞)→Rn for which there are two constantss1 >0,s2>0 such that
|d(t)| ≤s1e−s2t , ∀t≥0, (6.5) all the solutions of the system
˙
z(t) =Az(t) +Bus(z(t) +d(t)) (6.6) converge asymptotically to the origin.
Assumption 6.3. The unknown disturbance δ1 is piecewise continuous and such that, for allt≥0,
δ1(t)≤δ1(t)≤δ1(t), (6.7)
where δ1 : [0,+∞) → Rn and δ1 : [0,+∞) → Rn are known continuous functions. The unknown disturbance δ2 is such that, for all integer i∈N,
δ2,i≤δ2,i≤δ2,i, (6.8)
whereδ2,i,δ2,i are known constants.
Let us introduce some notation. LetDA∈Rn×ndenote the diagonal matrix such that all the diagonal entries ofA−DAare equal to zero. LetMA=DA+(A−DA)+,PA= (A−DA)−,
TA=
"
MA PA
PA MA
#
, QA=
"
MA −PA
−PA MA
#
, (6.9)
N =P−1, W =P L, ZN =
"
N+ −N−
−N− N+
#
(6.10) and, for all j∈N,j≥1,
Ra,j(∆∗) = Z tj
tj−1
h=+j (`)δ1(`)− =−j(`)δ1(`)i d` , Rb,j(∆∗) =
Z tj
tj−1
h
=+j (`)δ1(`)− =−j (`)δ1(`)i d` ,
(6.11)
with ∆∗ = (δ1, δ1), =j(`) = P J e(tj−`)A where J is the matrix defined in (6.4). Let
∆= (δ1, δ1, δ2, δ2) and, for allk∈N,k≥1,
S1,k(∆) =N[Ra,k(∆∗)−W+δ2,k+W−δ2,k],
S2,k(∆) =N[Rb,k(∆∗)−W+δ2,k+W−δ2,k]. (6.12) Notice for later use that there is a constant s3 > 0 such that, fori = 1,2, and all k∈ N, k≥1,
|Si,k(∆)| ≤s3
"
sup
`∈[tk−1,tk]
|∆∗(`)|+|δ2,k|+|δ2,k|
#
. (6.13)
Discussion of Theorem 6.4.
• If the discrete-time system gk+1 = eνAgk with the output Cgk is detectable, there is a matrix L1 ∈ Rn×q such that the matrix eνA+L1C is Schur stable. Then necessarily the matrix e−νA[eνA+L1C]eνA is Schur stable. It follows that the matrix [In−LC]eνA with L= −e−νAL1 is Schur stable, which implies that the first part of Assumption 6.1 is satisfied. Detectability of the pair (eνA, C) is a mild condition and, for some pairs (A, C), as for instance
"
0 1 0 0
#
,[1 0]
!
, it is satisfied for all ν > 0. We deduce that our as- sumptions imply less stringent requirements on the size of ν than the assumptions in [98].
Moreover, it is worth noticing that if the continuous-time system h˙ =Ahwith the output Ch is detectable, then there exists ν∗ > 0 such that for all ν ∈ (0, ν∗], the discrete-time system gk+1 =eνAgk with the output Cgk is detectable.
• For the sake of simplicity, the matrix P in Assumption 6.1 is constant. However, we conjecture that this assumption can be relaxed by replacing the matrix P by a sequence of matrices that can be deduced from the results in [95]. Then the corresponding interval observer (6.14)-(6.15) would be time-varying. If the pair (eνA, C) is observable, distinct
eigenvalues for the matrix [In−LC]eνA can be selected and then one can determine a constant matrixP for which Assumption 6.1is satisfied.
•Thexaandxb-subsystems of (6.14) are classical continuous-discrete observers for the sys- tem (6.2), which belong to the family of continuous-discrete observers used in [5]. Therefore the interval observer inherits the performances of the classical continuous-discrete observers.
•The goal of the xa andxb-subsystems of (6.14) is to provide with bounds for the solution x at the discrete instants tk, while the goal of the x and x-subsystems is to provide with bounds for the solution x over the intervals (tk, tk+1). An alternative choice of dynamics giving bounds over the intervals(tk, tk+1) is proposed in [93].
• Other choices of stabilizing feedback than (6.19) can be made. Among them, there is in particular u(xb) = us(xb). Observe that when each eigenvalue of A has a nonpositive real part, then Assumption6.2 is satisfied with bounded feedbacks of arbitrary size (see [130]).
Observe also that Assumption6.2 implies that the pair(A, B) is stabilizable, which implies that there is a matrixK ∈Rp×m such thatA+BK is Hurwitz. It follows that Assumption 6.2 is satisfied with the linear feedback us(z) =Kz. From this linear property and (6.18), one can easily deduce that this specific choice of feedback results in a closed-loop system possessing an ISS property with respect to δ1 andδ2.
We are ready to state and prove the following result:
Theorem 6.4. Assume that the system (6.2) satisfies Assumptions 6.1 to 6.3. Then the system defined, for allk∈N, by
˙ xa(t)
˙ xb(t)
!
= Axa(t) +Bu(t) Axb(t) +Bu(t)
!
, ∀t∈Ik, xa,k
xb,k
!
= xsa,k xsb,k
!
+ L(yk−Cxsa,k) L(yk−Cxsb,k)
!
+ S1,k(∆) S2,k(∆)
!
whenk≥1, x(t)˙
˙ x(t)
!
= QA
x(t) x(t)
!
+ Bu(t) +δ1(t) Bu(t) +δ1(t)
!
, ∀t∈Ik, xk
xk
!
= ZN P xa,k P xb,k
!
, whenk≥1,
(6.14) with the initial conditions
xa(0) xb(0) xa,0 xb,0
x0
x0
=
N[P+x0−P−x0] N[P+x0−P−x0] N[P+x0−P−x0] N[P+x0−P−x0]
x0
x0
(6.15)
and the boundsx,xis an interval observer for the system (6.2): i.e. whenu(t)is a piecewise continuous function defined over[0,+∞) and bounded over every compact set and
x0 ≤x(0)≤x0 (6.16)
then, for allt≥0,
x(t)≤x(t)≤x(t) (6.17)
and there are constants ci >0,i= 1 to 4such that, for all k∈N,k≥1,t∈Ik,
|x(t)−x(t)| ≤ c2e−c1t|x0−x0|+c3 sup
`∈[t0,tk]
|∆∗(`)|+c4 sup
i∈{1,...,k}
|δ2,i|+|δ2,i| . (6.18) Moreover all the solutions of the system (6.14)-(6.2) in closed-loop with the feedback
u(xa) =us(xa), (6.19)
whereusis the feedback given by Assumption6.2, converge to the origin when δ1 =δ1 = 0, δ2 =δ2= 0.
Proof. First step: existence and uniqueness of the solutions.
We consider a solution of (6.2)-(6.14) with the initial conditions selected in (6.15) under the assumption that u(t) is piecewise continuous, defined over [0,+∞) and bounded over every compact interval. One can show that such initial conditions generate one and only one solution of (6.14) as follows. From (6.15), it follows that the xa and xb-subsystems admit one and only one solution over[t0, t1). Moreover,xsa,1 and xsb,1 exist because u is piecewise continuous. On the other hand, from (6.15) we can also deduce that x and x-subsystems admit one and only one solution over [t0, t1). Next, arguing similarly over each interval Ik, one establishes the existence of one and only one solution over[0,+∞).
Second step: framer.
In this part of the proof, we show that (6.14) is a framer for (6.2).
Sincex0≤x0 ≤x0,P+≥0and P−≥0, by LemmaA.5in Appendix A, it follows that P+x0−P−x0 ≤P x(0)≤P+x0−P−x0 . (6.20) Using (6.15), we obtain
P xb,0≤P x(0)≤P xa,0 . (6.21) Now, to ease the analysis, we define two variables
ea=xa−x , eb =x−xb . (6.22)
They satisfy, for allk∈N,
˙
ea = Aea−δ1(t), ∀t∈Ik
ea,k = esa,k−LCesa,k+Lδ2,k+S1,k(∆), k≥1
˙
eb = Aeb+δ1(t), ∀t∈Ik
eb,k = esb,k−LCesb,k−Lδ2,k−S2,k(∆), k≥1.
(6.23)
By integrating, for any k∈N anyt∈Ik over[tk, t), we deduce that, for all k∈N, for all t∈Ik,
ea(t) =eA(t−tk)ea,k− Z t
tk
e(t−`)Aδ1(`)d`, (6.24) which implies that
esa,k+1 =eνAea,k− Z tk+1
tk
e(tk+1−`)Aδ1(`)d`.
Similarly, one can prove that
esb,k+1 =eνAeb,k+ Z tk+1
tk
e(tk+1−`)Aδ1(`)d` . These equalities and (6.23) give, for all k∈N,
ea,k+1 = Gea,k−J Z tk+1
tk
e(tk+1−`)Aδ1(`)d`+S1,k+1(∆) +Lδ2,k+1, eb,k+1 = Geb,k+J
Z tk+1
tk
e(tk+1−`)Aδ1(`)d`−S2,k+1(∆)−Lδ2,k+1 ,
(6.25)
whereG is the matrix defined in (6.3).
From the definitions of =j,N,W,S1,k,S2,k, andGin 6.11,6.10,6.12and in Assumption 6.1, we deduce that, for allk∈N,
P ea,k+1 = GP ea,k+Ra,k+1(∆∗)− Z tk+1
tk
=k+1(`)δ1(`)d`
+W δ2,k+1−W+δ2,k+1+W−δ2,k+1 , P eb,k+1 = GP eb,k−Rb,k+1(∆∗) +
Z tk+1
tk
=k+1(`)δ1(`)d`
−W δ2,k+1+W+δ2,k+1−W−δ2,k+1 .
(6.26)
Using the inequalities
(=k+1(`))+δ1(`)≤(=k+1(`))+δ1(`), (=k+1(`))+δ1(`)≤(=k+1(`))+δ1(`),
−(=k+1(`))−δ1(`)≤ −(=k+1(`))−δ1(`)
and
−(=k+1(`))−δ1(`)≤ −(=k+1(`))−δ1(`), we deduce that
Ra,k+1(∆∗)− Z tk+1
tk
=k+1(`)δ1(`)d`≥0 (6.27) and
Z tk+1
tk
=k+1(`)δ1(`)d`−Rb,k+1(∆∗)≥0 . (6.28) Moreover, using the Lemma A.5in AppendixA, we deduce that
W δ2,k+1−W+δ2,k+1+W−δ2,k+1 ≥0 (6.29) and
−W δ2,k+1+W+δ2,k+1−W−δ2,k+1 ≥0. (6.30) From (6.26), the inequalities (6.27), (6.28), (6.29), (6.30), the inequality G ≥ 0 and the fact that the inequalities in (6.21) are equivalent to
0≤P ea,0 , 0≤P eb,0, (6.31)
it follows that, for allk∈N,
0≤P eb,k , 0≤P ea,k , (6.32)
we deduce that, for allk∈N,
P xb,k≤P xk≤P xa,k . (6.33)
SinceN+ ≥0and N−≥0 and by LemmaA.5in Appendix A, we have, for allk∈N, N+P xb,k−N−P xa,k≤N P xk≤N+P xa,k−N−P xb,k (6.34) We deduce that, for allk∈N,
xk≤xk≤xk . (6.35)
Next, let us analyze x,x, andx over an interval Ik. SinceA=MA−PA, we have, for all t∈Ik,
( x˙ −x˙ = MA(x−x) +PA(x−x) +δ1(t)−δ1(t),
˙
x−x˙ = PA(x−x) +MA(x−x) +δ1(t)−δ1(t). (6.36)
Since, according to (6.35), for all k∈ N, x(tk)−x(tk)≥ 0,x(tk)−x(tk) ≥0 and for all t≥0,δ1(t)−δ1(t)≥0,δ1(t)−δ1(t)≥0, we deduce that
x(t)≤x(t)≤x(t) (6.37)
for all t ∈ Ik because the matrix TA defined in (6.9) is cooperative. This allows us to conclude.
Third step: stability analysis of the framer.
Lete˜=xa−xb. Then, for allk∈N,
( e(t)˙˜ = A˜e(t),∀t∈Ik,
˜
ek = Je˜sk+S3,k(∆), whenk≥1, (6.38) with
S3,k(∆) =S1,k(∆)−S2,k(∆). (6.39) By adding the two equations in (6.25), for allk≥0,
˜
ek+1 = G˜ek+S3,k+1(∆). (6.40) From Assumption6.1, we deduce that there exist real numbersri>0,i= 1 to3such that, for all k∈N,k≥1,
|˜ek| ≤r1e−kr2|˜e0|+r3 sup
j∈{1,...,k}
|S3,j(∆)|. (6.41)
By integrating the differential equation in (6.38), we deduce that for allt∈Ik,
|˜e(t)| ≤eν|A||˜e(tk)| ≤r1eν|A|e−kr2|˜e0|+eν|A|r3 sup
j∈{1,...,k}
|S3,j(∆)|. (6.42)
Now, observe that (6.14) implies that, whenk≥1, xk
xk
!
=ZN
P xb,k
P xb,k
! +ZN
Pe˜k
0
!
. (6.43)
It follows that, for all k≥1,
xk−xk= (N++N−)Pe˜k. (6.44) We deduce that there is a constant r4 >0 such that, for all t∈Ik,
|x(t)−x(t)| ≤r4|˜ek|. (6.45)
Bearing in mind (6.41), we deduce that there are constants r5 >0,r6 >0 such that
|x(t)−x(t)| ≤r5e−kr2|˜e0|+r6 sup
j∈{1,...,k}
|S3,j(∆)|. (6.46) Bearing in mind (6.13), we deduce that (6.18) is satisfied.
Fourth step: stability analysis of the closed-loop system.
Let us establish that all the solutions of the system (6.14)-(6.2) in closed-loop with the feedback in (6.19) converge to the origin when δ1 =δ1= 0,δ2=δ2= 0.
Observe that, for all t≥0,
˙
x(t) =Ax(t) +us(x(t) +ea(t)). (6.47) We deduce from (6.25) that there are constants r7 >0andr8>0 such that, for allk∈N,
|ea(tk)| ≤r7e−r8k|˜e0|. (6.48) We deduce that there is a constant r9 >0 such that for allt≥0,
|ea(t)| ≤r9e−r8t|˜e0|. (6.49) This inequality, (6.47) and Assumption6.2allow us to conclude.
6.4 Example
Throughout this section, we use the notation of Section6.3 and denote by vi the ith com- ponent of the vectorv. To illustrate Theorem 6.4, we consider the two dimensional system of the family (6.2) with the matrices:
A=
"
0 1
−1 0
# , B =
"
0 1
#
, C = [1 0], (6.50)
with disturbances admitting the bounds δ1 =c[1 1]>,δ1 =−δ1,δ2,k =c, δ2,k =−δ2,k, wherec is a constant.
Next, we construct a continuous-discrete interval observer. Let us choose
L= [1 1]>. (6.51)
ThenJ =
"
0 0
−1 1
#
, eνA=
"
cos(ν) sin(ν)
−sin(ν) cos(ν)
#
,which implies that the choiceν= π4 gives the matrix
G=−
"
0 0
√ 2 0
#
, (6.52)
which is Schur stable, but is not nonnegative. One can check readily that Assumption6.1is satisfied withP =
"
1 0 0 −1
#
sinceG=P GP−1 =−G. Since Assumption6.2 is satisfied with, for instance,
us(x) =−5
2x2, (6.53)
both Assumptions 1 and 2 are satisfied. Therefore, Theorem 6.4 applies. Then, through simple calculations, we obtain that the system defined, for allk∈N, by
˙
xa(t) = Axa(t) +Bu(t), ∀t∈Ik
˙
xb(t) = Axb(t) +Bu(t), ∀t∈Ik
xa,k = yk+c
yk+xs2,a,k−xs1,a,k−(1 +√ 2)c
!
, when k≥1
xb,k = yk−c
yk+xs2,b,k−xs1,b,k+ (1 +√ 2)c
!
, when k≥1 x(t)˙ = x2(t) +c
−x1(t) +c+u(t)
!
, ∀t∈Ik
˙
x(t) = x2(t)−c
−x1(t)−c+u(t)
!
, ∀t∈Ik xk = x1,a,k
x2,b,k
!
, xk= x1,b,k x2,a,k
!
(6.54)
with yk=x1(tk) and the initial conditions
xa(0) =xa,0= x1,0 x2,0
!
, (6.55)
xb(0) =xb,0 = x1,0 x2,0
!
(6.56) such that
x0 ≤x(0)≤x0 (6.57)
and the boundsx,x is an interval observer for the considered system.
We present simulations for the system (6.50)-(6.54) with us defined in (6.53) as feedback.
0 1 2 3 4 5 6
−5 0 5 10 15 20
Time
x2
Figure 6.1: Evolution of the state componentx2 and its bounds without uncertainty.
1 2 3 4 5 6 7
−5 0 5 10 15 20
Time
x2
Figure 6.2: Evolution of the state componentx2and its bounds with the uncertainties.
6.5 Conclusion
We have developed a new technique of construction of continuous-discrete interval observers for continuous-time systems with discrete measurements and disturbances in the measure- ments and the dynamics.
Many extensions of this result are possible. We plan to investigate the case where the sequence of the differences between two consecutive instants at which the measurements are available is not constant and to design reduced order interval observers in the spirit of what is done in [40].
Moreover, nonlinear systems with globally Lipschitz nonlinearities or triangular structures, time-varying systems and systems with delay (notably in the input) may be considered.
Conclusions and Perspectives
Contents
7.1 Contributions of this dissertation . . . . 123 7.2 Future works . . . . 125
7.1 Contributions of this dissertation
This thesis presents new results in the field of state estimation based on the theory of positive systems. It is composed of two separate parts. The first one studies the problem of positive observer design for positive systems. The second one which deals with robust state estimation through the design of interval observers, is at the core of our work.
We began our thesis by proposing the design of a nonlinear positive observer for discrete-time positive time-varying linear systems based on the use of generalized polar coordinates in the positive orthant. The idea underlying the method is that first, the direction of the true state is correctly estimated in the projective space thanks to the Birkhoff theorem and then very mild assumptions on the output map allow to reconstruct the norm of the state.
Later, the thesis was continued by studying the so-called interval observers for different families of dynamic systems in continuous-time, in discrete-time and also in a context
”continuous-discrete” (i.e. a class of continuous-time systems with discrete-time measure- ments). Thanks to interval observers, one can construct control laws which stabilize the considered systems. Precisely, five families of systems were mentioned in this thesis :
• A family of nonlinear continuous-time systems affine in the unmeasured part of state variables.
• A family of nonlinear discrete-time systems which possess specific stability and monotonic- ity properties.
123
• A fundamental family of linear time-invariant discrete-time systems.
• A family of nonlinear discrete-time systems with input and output.
• A family of continuous-time systems with discrete measurements.
Here we outline the key contributions of the dissertation as follows.
First, for a family of continuous-time nonlinear systems with input, output and uncertain terms affine in the unmeasured state components, a new interval observer design has been proposed. The main feature of the constructed interval observer is that it is composed of two copies of a classical observer whose corresponding error equations are, in general, not cooperative. The interval observer is globally asymptotically stable under an appropriate choice of dynamic output feedback which uses the values of the output and the bounds provided by the interval observer itself.
Secondly, discrete-time systems are very important and play a large role from a theoretical as well as an applied point of view (note the fact that discretization techniques transform continuous-time systems into discrete-time systems). Therefore, this kind of systems became the next subject of our study. First, time-invariant interval observers have been proposed for a family of nonlinear discrete-time systems which possesses specific stability and mono- tonicity properties. Second, it has been shown that, for any time-invariant exponentially stable discrete-time linear system with additive disturbances, time-varying exponentially sta- ble discrete-time interval observers can be constructed. The latter result relies on the design of time-varying changes of coordinates which transform a linear system into a nonnegative one.
In order to complement the previous results, a new interval observer has been designed for a family of nonlinear discrete-time systems with input, output and uncertain terms. Its main feature is that, as presented in the case of nonlinear continuous-time systems, it is composed of two copies of classical observers. This interval observer has applied in the presence of unknown bounded nonlinear terms and additive disturbances and has also been used to achieve asymptotic stability through an appropriate choice of dynamic output feedback.
After constructing interval observers for both continuous-time and discrete-time systems, a question about interval observer design for continuous-time systems with discrete measure- ments has naturally arisen and the last chapter of the dissertation was devoted to solving this problem. Here we have considered continuous-time systems with input, output and additive disturbances in the particular case where the measurements are only available at discrete instants and have disturbances. For these systems, continuous-discrete interval ob- servers that are asymptotically stable in the absence of disturbances have been constructed.
These interval observers are composed of two copies of the studied system and of a framer,
accompanied with appropriate outputs which give, componentwise, upper and lower bounds for the solutions of the studied system.
7.2 Future works
Positive observer
In the last ten years or so, the design of positive observers for positive systems (i.e. observers such that the estimated state respects positivity) has attracted an ever growing attention.
Indeed, the requirement that the observer estimates be positive at any time seems desirable since it allows a physical interpretation at all times, that is, even when the estimation error is not small. The Chapter 2 gives some ideas how the Hilbert metric is useful to analyse contraction and convergence properties of positive maps. For future research, one can extend these results to the nonlinear case and to more general cones. For example, the cone of hermitian positive definite matrices on which the Hilbert metric is presented as follow :
d(X, Y) = log λmax XY−1 λmin(XY−1)
! .
Interval observer
In spite of the results already available in the literature, interval observers are fertile ground for further studies. One of many research directions can be mentioned here, for example systems with delay. Extensions of the results presented in Chapter 4 to families of linear discrete-time systems with a point-wise delay or with delay in the input, families of discrete- time nonlinear systems with delay can be expected.
Moreover, we can also extend the results of Chapter 3 to systems with other special struc- tures, such as nonlinear systems with globally Lipschitz nonlinearities or triangular systems.
Families of time-varying systems may be considered as well. Extensions of the results in Chapter5 to discrete-time time-varying systems may be the subject of further research.
Furthermore, design of interval observers for continuous-time systems with discrete mea- surements is another beautiful problem to solve. For example, extensions of the results in Chapter6to the case where the sequence of the differences between two consecutive instants at which the measurements are available is not constant can be investigated. Besides, design reduced order interval observers in the spirit of what is done in [40] can be expected too.
Technical Lemmas
We prove the lemmas needed in several chapters.
Lemma A.1. Let
M=
"
−κ ω
−ω −κ
#
(A.1) whereκandω are two real numbers such thatκ2+ω2>0. Letαbe any real number such that
sin(α) =−√ ω
κ2+ω2 , cos(α) =−√ κ
κ2+ω2 (A.2)
and let, for allj∈N,
Lj =
"
cos(αj) sin(αj)
−sin(αj) cos(αj)
#
. (A.3)
Then, for allk∈N, the equality
Lk+1ML−1k =p
κ2+ω2I2 (A.4)
is satisfied.
Proof. The equalities
sin(α(k+ 1)) = sin(αk) cos(α) + cos(αk) sin(α) ,
cos(α(k+ 1)) = cos(αk) cos(α)−sin(αk) sin(α) , (A.5)
127
imply that
Lk+1 = √ 1
κ2+ω2
"
−κcos(αk) +ωsin(αk) −κsin(αk)−ωcos(αk) κsin(αk) +ωcos(αk) −κcos(αk) +ωsin(αk)
#
= √ 1
κ2+ω2
"
cos(αk) sin(αk)
−sin(αk) cos(αk)
# "
−κ −ω
ω −κ
# .
(A.6)
Therefore
Lk+1 = √ 1
κ2+ω2LkM> . (A.7)
From the equality
M>M = κ2+ω2
I2 (A.8)
we deduce that
Lk+1M = √
κ2+ω2Lk . (A.9)
This allows us to conclude.
Lemma A.2. We consider the system
ak+1 =Jak , k∈N (A.10)
with
J =
M I2 0 . . . 0 0 M I2 . .. ...
... . .. ... ... 0 ... . .. ... I2 0 . . . 0 M
∈R2p×2p, (A.11)
withMdefined in (A.1). Then there exists a constantα such that the time-varying change of coordinates
bk=Lkak (A.12)
with
Lk=diag{Lk−p+1,Lk−p+2, ...,Lk−1,Lk} ∈R2p×2p (A.13) with, for all integerj,Lj defined in (A.3), transforms the system (A.10) into the system
bk+1=p
κ2+ω2bk+Kbk, (A.14)
with
K =
0 I2 0 . . . 0 0 0 I2 . .. ...
... . .. ... ... 0 ... . .. ... I2 0 . . . 0 0
∈R2p×2p. (A.15)
Proof. Letα be any real number such that sin(α) = −√ ω
κ2+ω2 , cos(α) = −√ κ
κ2+ω2 . (A.16)
According to the definition ofbk in (A.12) and (A.10),
bk+1=Lk+1Jak=Lk+1J L−1k bk=Lk+1J L>kbk . (A.17) One can check readily that
J L>k =
ML>k−p+1 L>k−p+2 0 . . . 0 0 ML>k−p+2 L>k−p+3 . .. ...
... . .. . .. . .. 0 ... . .. . .. L>k 0 . . . 0 ML>k
. (A.18)
Now, from (A.9), we deduce that
Lk+1= 1
√
κ2+ω2
Lk−p+1M> 0 . . . 0
0 Lk−p+2M> 0 . . . 0
... . .. ...
0 . . . 0 Lk−1M> 0 0 . . . 0 LkM>
. (A.19)
It follows that
Lk+1J L>k = 1
√
κ2+ω2
ϕk,1 νk,1 0 . . . 0 0 ϕk,2 νk,2 . .. ... ... . .. . .. ... 0 ... . .. ... νk,p−1
0 . . . 0 ϕk,p
(A.20)
with
ϕk,i=Lk−p+iM>ML>k−p+i (A.21)
and
νk,i=Lk−p+iM>L>k−p+i+1. (A.22) SinceM>M= (κ2+ω2)I2 andLiL>i =I2, we deduce that
ϕk,i= (κ2+ω2)Lk−p+iL>k−p+i = (κ2+ω2)I2 . (A.23) Now, observe that (A.9) implies that
LiM>L>i+1=LiM> 1
√
κ2+ω2ML>i . (A.24) Using again the equalitiesM>M= (κ2+ω2)I2 andLiL>i =I2, we obtain
LiM>L>i+1=p
κ2+ω2LiL>i =p
κ2+ω2I2 . (A.25) We deduce that
Lk+1J L>k =
√
κ2+ω2I2 I2 0 . . . 0
0 √
κ2+ω2I2 I2 . .. ...
... . .. . .. ... 0
... . .. ... I2
0 . . . 0 √
κ2+ω2I2
. (A.26)
This allows to conclude.
Lemma A.3. We consider the system
ak+1 =Jaak , k∈N (A.27)
with
Ja =
−µ 1 . . . 0 0 −µ . .. ...
... . .. ... 1 0 . . . 0 −µ
∈Rn×n, (A.28)
whereµ is a positive real number. Then, the time-varying change of coordinates
bk=Gkak (A.29)
with
Gk=diag{(−1)k,(−1)k+1, ...,(−1)k+n−1} ∈Rn×n (A.30)
gives
bk+1=Jbbk , k∈N (A.31)
with
Jb=
µ 1 . . . 0 0 µ . .. ...
... . .. ... 1 0 . . . 0 µ
∈Rn×n. (A.32)
Proof. According to (A.29) and (A.27),
bk+1 = Gk+1Jaak = Gk+1JaGk−1bk . (A.33) SinceGk−1 =Gk andGk+1=−Gk, the sequencebk satisfies
bk+1 =−GkJaGkbk . (A.34)
Now, we have
JaGk=
µ(−1)k+1 (−1)k+1 . . . 0 0 µ(−1)k+2 . .. ...
... . .. . .. (−1)k+n−1 0 . . . 0 µ(−1)k+n
. (A.35)
Therefore
GkJaGk = −
µ 1 . . . 0 0 µ . .. ...
... . .. ... 1 0 . . . 0 µ
. (A.36)
This equality and (A.34) allow to conclude.
Proof of Theorem 4.9. Step 1: Jordan canonical forms.
From [110, Section 1.8], we deduce that for some integersr ∈ {0,1, ..., n},s∈ {0,1, ..., n− 1} there exists a linear time-invariant change of coordinates
gk=Pxk, (A.37)
which transforms (4.47) into
gk+1=Jgk, (A.38)
with
J =diag{J1,J2, ...,Js} ∈Rn×n, (A.39) where the matrices Ji are partitioned into two groups: the first r matrices are associated with the r real eigenvalues of multiplicity ni of A and the others are associated with the imaginary eigenvalues of multiplicity mi of A. Therefore n =
r
X
i=1
ni+
s
X
r+1
2mi and, for i= 1 to r,
Ji =
−µi 1 . . . 0 0 −µi . .. ...
... . .. ... 1 0 . . . 0 −µi
∈Rni×ni, (A.40)
where theµi’s are real numbers and, fori=r+ 1to s,
Ji =
Mi I2 0 . . . 0 0 Mi I2 . .. ...
... . .. ... ... 0 ... . .. ... I2
0 . . . 0 Mi
∈R2mi×2mi, (A.41)
with
Mi =
"
−κi ωi
−ωi −κi
#
∈R2×2, (A.42)
and
I2 =diag{1,1} ∈R2×2, (A.43)
where theωi’s are non-zero real numbers.
Step 2: time-varying change of coordinates. We consider the system (A.38). From Lemmas A.2andA.3in Appendix A, we deduce that, for any system
ak+1=Jiak, (A.44)
there exist a nonnegative matrix Hi with a spectral radius smaller than 1 and a sequence (Qk,i) of invertible matrices bounded in norm by1 and with inverses bounded in norm by 1 such that the change of coordinates
bk=Qk,iak (A.45)
gives
bk+1 =Hibk. (A.46)
Next, we consider the change of coordinates
hk =Qkgk, (A.47)
with
Qk =diag{Qk,1,Qk,2, ...,Qk,s} ∈Rn×n. (A.48) Then
hk+1=Qk+1Jgk =Qk+1J Q−1k hk=Hhk (A.49) with
H=diag{H1,H2, ...,Hs} ∈Rn×n. Finally, we conclude by observing that the change of coordinates
hk =Rkxk, (A.50)
with Rk=QkP gives the nonnegative exponentially stable time-invariant system hk+1 =Hhk
and thatR−1k =P−1Q−1k , which implies that the sequences (Rk) and(R−1k ) are bounded in norm.
Lemma A.4. Let f :Rm→R be a function of classC1. Then there exists a functionfc: Rm×Rm →Rnondecreasing with respect to each of itsm first variables and nonincreasing with respect to each of its m last variables such that, for allx∈Rm, the equality
fc(x,x) =f(x) (A.51)
is satisfied.
Proof. The function
Γ(s) = sup
|z|≤√ s
F(z) + 1 +s, (A.52)
whereF :Rm →Ris defined by
F(x) =|x||f(x)|+
∂f
∂x(x)
, (A.53)
is continuous, positive and nondecreasing over [0,+∞). Then the function ξ(s) = 1
2 Z +∞
s
1
emΓ(m)dm (A.54)
is well-defined, decreasing, continuously differentiable, such that, for alls≥0,ξ(s)>0and ξ(s)≤ 1
2Γ(s) Z +∞
s
1
emdm≤ 1
2Γ(s), (A.55)
|ξ0(s)| ≤ 1
2Γ(s). (A.56)
Now, we observe that, for allx∈Rm,
f(x) =f1(x) +f2(x) (A.57) with
f1(x) =
4 m
X
j=1
xj
ξ(|x|2) , f2(x) = ξ(|x|f3(x)2) , f3(x) =ξ(|x|2)f(x)−4
m
X
j=1
xj. (A.58)
Moreover, (A.54) and (A.55) imply that, for alli∈ {1, ..., m}and for allx∈Rm,
∂f3
∂xi(x) = 2ξ0(|x|2)xif(x) +ξ(|x|2)∂x∂f
i(x)−4
≤ 2|ξ0(|x|2)||xi||f(x)|+ξ(|x|2)
∂f
∂xi(x) −4
≤ Γ(|x|12)|xi||f(x)|+2Γ(|x|1 2)
∂f
∂xi(x) −4
≤ Γ(|x|F(x)2) −4
≤ −3,
(A.59)
where the last inequality is deduced from the definition of Γ. Therefore f3 is nonincreasing with respect to each of its variables.
Now, we define two functions:
φ1(a, b) =
4 m
X
j=1
$(aj)
ξ
m
X
j=1
$(aj)2+
m
X
j=1
f(bj)2
+
4 m
X
j=1
f(aj)
ξ
m
X
j=1
$(bj)2+
m
X
j=1
f(aj)2
, (A.60)
φ2(a, b) = $(f3(b))
ξ
m
X
j=1
$(aj)2+
m
X
j=1
f(bj)2
+ f(f3(b))
ξ
m
X
j=1
$(bj)2+
m
X
j=1
f(aj)2
.
(A.61) Using the fact that, for allx∈Rm,
|x|2 =
m
X
j=1
x2j =
m
X
j=1
$(xj)2+
m
X
j=1
f(xj)2 (A.62)